7.3: Circle Equations

Front

What is the standard form of the equation of a circle?

Back

The standard form of the equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

Front

How do you find the center of a circle from its equation?

Back

The center of a circle in the equation (x - h)² + (y - k)² = r² is given by the point (h, k).

Front

What is the radius of a circle in the equation (x - 3)² + (y + 4)² = 16?

Back

The radius is 4 units, since r² = 16, thus r = √16 = 4.

Front

If the diameter of a circle is 10, what is the radius?

Back

The radius is 5 units, since the radius is half of the diameter.

Front

What are the coordinates of the center of the circle with the equation (x + 2)² + (y - 1)² = 25?

Back

The center is at (-2, 1).

Front

How do you convert the general form of a circle's equation to standard form?

Back

To convert from general form Ax² + Ay² + Bx + Cy + D = 0 to standard form, complete the square for x and y.

Front

What is the equation of a circle with center (0, 0) and radius 6?

Back

The equation is x² + y² = 36.

Front

What does the term 'completing the square' mean in the context of circle equations?

Back

Completing the square is a method used to rewrite a quadratic equation in standard form by transforming it into a perfect square trinomial.

Front

What is the significance of the radius in the equation of a circle?

Back

The radius determines the size of the circle; it is the distance from the center to any point on the circle.

Front

How can you determine if a point lies inside, on, or outside a circle?

Back

Substitute the point's coordinates into the circle's equation. If the result is less than r², it's inside; if equal, on; if greater, outside.